R0043-22184 · q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0059
Step-by-step derivation
S001 · implies
Before
source operation=mobinv-roundtrip-table;original snapshot and mathematics retained
After
explicit ordered source output fields=[n, roundtrip_rows, all_equal, boundary]
Conditions: none
Retain the historical interface marker and list ordered source fields without JavaScript.MVT fields come from the original mode,without padding or answer-shape inference.Tags and fixed No markers retain their claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
Mobius inversion ORIGINAL roundtrip:actual positive input=9;check complete integer range1..9
After
ALL positive divisors INCREASING=[1, 3, 9];not only primes/proper divisors or all integers<=n
Conditions: none
Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.
Step identifiers and dependencies
Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S001
S003 · implies
Before
Mobius roundtrip m=1 forward:actual positive input=1;check complete integer range1..1
After
ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n
Conditions: none
Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.
Step identifiers and dependencies
Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S002
S004 · calculation
Before
Mobius roundtrip m=1 forward f(1) ORIGINAL a times divisor := mul(4, 1)
After
Mobius roundtrip m=1 forward f(1) ORIGINAL a times divisor = 4
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · calculation
Before
Mobius roundtrip m=1 forward f(1) plus ORIGINAL b := sum(4, -4)
After
Mobius roundtrip m=1 forward f(1) plus ORIGINAL b = 0
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · calculation
Before
Mobius roundtrip m=1 forward full forward divisor sum := sum(0)
After
Mobius roundtrip m=1 forward full forward divisor sum = 0
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · implies
Before
Mobius roundtrip m=1 inverse:actual positive input=1;check complete integer range1..1
After
ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n
Conditions: none
Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.
Step identifiers and dependencies
Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S006
S008 · implies
Before
Mobius roundtrip m=1 inverse Mobius at ORIGINAL divisor1:actual positive input for mu=1
After
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
Conditions: none
Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search.
Step identifiers and dependencies
Rule: mobius_complete_UPF_input_role · Uses: G001, S007
S009 · implies
Before
Mobius roundtrip m=1 inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]
After
mu=1;empty product case=Yes;repeated prime=No
Conditions: none
The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.
Step identifiers and dependencies
Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S008
S010 · calculation
Before
Mobius roundtrip m=1 inverse n divided by ORIGINAL divisor1 := div(1, 1)
After
Mobius roundtrip m=1 inverse n divided by ORIGINAL divisor1 = 1
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · implies
Before
Mobius roundtrip m=1 inverse F(n/1):actual positive input=1;check complete integer range1..1
After
ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n
Conditions: none
Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.
Step identifiers and dependencies
Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S010
S012 · calculation
Before
Mobius roundtrip m=1 inverse F(n/1) f(1) ORIGINAL a times divisor := mul(4, 1)
After
Mobius roundtrip m=1 inverse F(n/1) f(1) ORIGINAL a times divisor = 4
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S011
S013 · calculation
Before
Mobius roundtrip m=1 inverse F(n/1) f(1) plus ORIGINAL b := sum(4, -4)
After
Mobius roundtrip m=1 inverse F(n/1) f(1) plus ORIGINAL b = 0
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S012
S014 · calculation
Before
Mobius roundtrip m=1 inverse F(n/1) full forward divisor sum := sum(0)
After
Mobius roundtrip m=1 inverse F(n/1) full forward divisor sum = 0
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S013
S015 · calculation
Before
Mobius roundtrip m=1 inverse mu(1) times SAME F(n/1) := mul(1, 0)
After
Mobius roundtrip m=1 inverse mu(1) times SAME F(n/1) = 0
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S014
S016 · calculation
Before
Mobius roundtrip m=1 inverse complete inverse-term sum := sum(0)
After
Mobius roundtrip m=1 inverse complete inverse-term sum = 0
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S015
S017 · calculation
Before
Mobius roundtrip m=1 inverse direct ORIGINAL a times n := mul(4, 1)
After
Mobius roundtrip m=1 inverse direct ORIGINAL a times n = 4
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S016
S018 · calculation
Before
Mobius roundtrip m=1 inverse direct plus ORIGINAL b := sum(4, -4)
After
Mobius roundtrip m=1 inverse direct plus ORIGINAL b = 0
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S017
S019 · implies
Before
Mobius roundtrip m=1 inverse:actual n=1;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 1, 0, 0]]
After
complete reconstructed=0;direct f(n)=0;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)
Conditions: none
Swap the finite sums over d*e|n.The inner Mobius sum is1 for n/e=1 and0 otherwise by pairing distinct-prime subsets,leaving only e=n.For n1,the d1 term and mu1=1 remain;do not mix indices from the alternate equivalent convention.
Step identifiers and dependencies
Rule: mobinv_general_finite_double_sum_and_Mobius_cancellation · Uses: G001, S018
S020 · implies
Before
Mobius roundtrip m=3 forward:actual positive input=3;check complete integer range1..3
After
ALL positive divisors INCREASING=[1, 3];not only primes/proper divisors or all integers<=n
Conditions: none
Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.
Step identifiers and dependencies
Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S019
S021 · calculation
Before
Mobius roundtrip m=3 forward f(1) ORIGINAL a times divisor := mul(4, 1)
After
Mobius roundtrip m=3 forward f(1) ORIGINAL a times divisor = 4
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S020
S022 · calculation
Before
Mobius roundtrip m=3 forward f(1) plus ORIGINAL b := sum(4, -4)
After
Mobius roundtrip m=3 forward f(1) plus ORIGINAL b = 0
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S021
S023 · calculation
Before
Mobius roundtrip m=3 forward f(3) ORIGINAL a times divisor := mul(4, 3)
After
Mobius roundtrip m=3 forward f(3) ORIGINAL a times divisor = 12
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S022
S024 · calculation
Before
Mobius roundtrip m=3 forward f(3) plus ORIGINAL b := sum(12, -4)
After
Mobius roundtrip m=3 forward f(3) plus ORIGINAL b = 8
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S023
S025 · calculation
Before
Mobius roundtrip m=3 forward full forward divisor sum := sum(0, 8)
After
Mobius roundtrip m=3 forward full forward divisor sum = 8
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S024
S026 · implies
Before
Mobius roundtrip m=3 inverse:actual positive input=3;check complete integer range1..3
After
ALL positive divisors INCREASING=[1, 3];not only primes/proper divisors or all integers<=n
Conditions: none
Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.
Step identifiers and dependencies
Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S025
S027 · implies
Before
Mobius roundtrip m=3 inverse Mobius at ORIGINAL divisor1:actual positive input for mu=1
After
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
Conditions: none
Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search.
Step identifiers and dependencies
Rule: mobius_complete_UPF_input_role · Uses: G001, S026
S028 · implies
Before
Mobius roundtrip m=3 inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]
After
mu=1;empty product case=Yes;repeated prime=No
Conditions: none
The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.
Step identifiers and dependencies
Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S027
S029 · calculation
Before
Mobius roundtrip m=3 inverse n divided by ORIGINAL divisor1 := div(3, 1)
After
Mobius roundtrip m=3 inverse n divided by ORIGINAL divisor1 = 3
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S028
S030 · implies
Before
Mobius roundtrip m=3 inverse F(n/1):actual positive input=3;check complete integer range1..3
After
ALL positive divisors INCREASING=[1, 3];not only primes/proper divisors or all integers<=n
Conditions: none
Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.
Step identifiers and dependencies
Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S029
S031 · calculation
Before
Mobius roundtrip m=3 inverse F(n/1) f(1) ORIGINAL a times divisor := mul(4, 1)
After
Mobius roundtrip m=3 inverse F(n/1) f(1) ORIGINAL a times divisor = 4
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S030
S032 · calculation
Before
Mobius roundtrip m=3 inverse F(n/1) f(1) plus ORIGINAL b := sum(4, -4)
After
Mobius roundtrip m=3 inverse F(n/1) f(1) plus ORIGINAL b = 0
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S031
S033 · calculation
Before
Mobius roundtrip m=3 inverse F(n/1) f(3) ORIGINAL a times divisor := mul(4, 3)
After
Mobius roundtrip m=3 inverse F(n/1) f(3) ORIGINAL a times divisor = 12
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S032
S034 · calculation
Before
Mobius roundtrip m=3 inverse F(n/1) f(3) plus ORIGINAL b := sum(12, -4)
After
Mobius roundtrip m=3 inverse F(n/1) f(3) plus ORIGINAL b = 8
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S033
S035 · calculation
Before
Mobius roundtrip m=3 inverse F(n/1) full forward divisor sum := sum(0, 8)
After
Mobius roundtrip m=3 inverse F(n/1) full forward divisor sum = 8
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S034
S036 · calculation
Before
Mobius roundtrip m=3 inverse mu(1) times SAME F(n/1) := mul(1, 8)
After
Mobius roundtrip m=3 inverse mu(1) times SAME F(n/1) = 8
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S035
S037 · implies
Before
Mobius roundtrip m=3 inverse Mobius at ORIGINAL divisor3:actual positive input for mu=3
After
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
Conditions: none
Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search.
Step identifiers and dependencies
Rule: mobius_complete_UPF_input_role · Uses: G001, S036
S038 · implies
Before
ORIGINAL positive integer absolute magnitude=3>1
After
initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted
Conditions: none
Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime.
Step identifiers and dependencies
Rule: upf_factorization_remaining_product_invariant · Uses: G001, S037
S039 · calculation
Before
upf canonical complete product factor0 := mul(1, 3)
After
upf canonical complete product factor0 = 3
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S038
S040 · implies
Before
exhausted trial divisors below2;terminal remaining=3;d*d>remaining;ORIGINAL n=3
After
complete increasing prime multiset=[3];full product=3;a composite terminal remainder would have an already tested smaller divisor
Conditions: none
If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors.
Step identifiers and dependencies
Rule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S039
S041 · implies
Before
Mobius roundtrip m=3 inverse Mobius at ORIGINAL divisor3:input=3;COMPLETE prime multiset=[3];distinct primes=[3]
After
mu=-1;empty product case=No;repeated prime=No
Conditions: none
The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.
Step identifiers and dependencies
Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S040
S042 · calculation
Before
Mobius roundtrip m=3 inverse n divided by ORIGINAL divisor3 := div(3, 3)
After
Mobius roundtrip m=3 inverse n divided by ORIGINAL divisor3 = 1
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S041
S043 · implies
Before
Mobius roundtrip m=3 inverse F(n/3):actual positive input=1;check complete integer range1..1
After
ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n
Conditions: none
Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.
Step identifiers and dependencies
Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S042
S044 · calculation
Before
Mobius roundtrip m=3 inverse F(n/3) f(1) ORIGINAL a times divisor := mul(4, 1)
After
Mobius roundtrip m=3 inverse F(n/3) f(1) ORIGINAL a times divisor = 4
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S043
S045 · calculation
Before
Mobius roundtrip m=3 inverse F(n/3) f(1) plus ORIGINAL b := sum(4, -4)
After
Mobius roundtrip m=3 inverse F(n/3) f(1) plus ORIGINAL b = 0
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S044
S046 · calculation
Before
Mobius roundtrip m=3 inverse F(n/3) full forward divisor sum := sum(0)
After
Mobius roundtrip m=3 inverse F(n/3) full forward divisor sum = 0
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S045
S047 · calculation
Before
Mobius roundtrip m=3 inverse mu(3) times SAME F(n/3) := mul(-1, 0)
After
Mobius roundtrip m=3 inverse mu(3) times SAME F(n/3) = 0
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S046
S048 · calculation
Before
Mobius roundtrip m=3 inverse complete inverse-term sum := sum(8, 0)
After
Mobius roundtrip m=3 inverse complete inverse-term sum = 8
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S047
S049 · calculation
Before
Mobius roundtrip m=3 inverse direct ORIGINAL a times n := mul(4, 3)
After
Mobius roundtrip m=3 inverse direct ORIGINAL a times n = 12
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S048
S050 · calculation
Before
Mobius roundtrip m=3 inverse direct plus ORIGINAL b := sum(12, -4)
After
Mobius roundtrip m=3 inverse direct plus ORIGINAL b = 8
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S049
S051 · implies
Before
Mobius roundtrip m=3 inverse:actual n=3;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 3, 8, 8], [3, -1, 1, 0, 0]]
After
complete reconstructed=8;direct f(n)=8;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)
Conditions: none
Swap the finite sums over d*e|n.The inner Mobius sum is1 for n/e=1 and0 otherwise by pairing distinct-prime subsets,leaving only e=n.For n1,the d1 term and mu1=1 remain;do not mix indices from the alternate equivalent convention.
Step identifiers and dependencies
Rule: mobinv_general_finite_double_sum_and_Mobius_cancellation · Uses: G001, S050
S052 · implies
Before
Mobius roundtrip m=9 forward:actual positive input=9;check complete integer range1..9
After
ALL positive divisors INCREASING=[1, 3, 9];not only primes/proper divisors or all integers<=n
Conditions: none
Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.
Step identifiers and dependencies
Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S051
S053 · calculation
Before
Mobius roundtrip m=9 forward f(1) ORIGINAL a times divisor := mul(4, 1)
After
Mobius roundtrip m=9 forward f(1) ORIGINAL a times divisor = 4
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S052
S054 · calculation
Before
Mobius roundtrip m=9 forward f(1) plus ORIGINAL b := sum(4, -4)
After
Mobius roundtrip m=9 forward f(1) plus ORIGINAL b = 0
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S053
S055 · calculation
Before
Mobius roundtrip m=9 forward f(3) ORIGINAL a times divisor := mul(4, 3)
After
Mobius roundtrip m=9 forward f(3) ORIGINAL a times divisor = 12
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S054
S056 · calculation
Before
Mobius roundtrip m=9 forward f(3) plus ORIGINAL b := sum(12, -4)
After
Mobius roundtrip m=9 forward f(3) plus ORIGINAL b = 8
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S055
S057 · calculation
Before
Mobius roundtrip m=9 forward f(9) ORIGINAL a times divisor := mul(4, 9)
After
Mobius roundtrip m=9 forward f(9) ORIGINAL a times divisor = 36
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S056
S058 · calculation
Before
Mobius roundtrip m=9 forward f(9) plus ORIGINAL b := sum(36, -4)
After
Mobius roundtrip m=9 forward f(9) plus ORIGINAL b = 32
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S057
S059 · calculation
Before
Mobius roundtrip m=9 forward full forward divisor sum := sum(0, 8, 32)
After
Mobius roundtrip m=9 forward full forward divisor sum = 40
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S058
S060 · implies
Before
Mobius roundtrip m=9 inverse:actual positive input=9;check complete integer range1..9
After
ALL positive divisors INCREASING=[1, 3, 9];not only primes/proper divisors or all integers<=n
Conditions: none
Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.
Step identifiers and dependencies
Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S059
S061 · implies
Before
Mobius roundtrip m=9 inverse Mobius at ORIGINAL divisor1:actual positive input for mu=1
After
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
Conditions: none
Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search.
Step identifiers and dependencies
Rule: mobius_complete_UPF_input_role · Uses: G001, S060
S062 · implies
Before
Mobius roundtrip m=9 inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]
After
mu=1;empty product case=Yes;repeated prime=No
Conditions: none
The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.
Step identifiers and dependencies
Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S061
S063 · calculation
Before
Mobius roundtrip m=9 inverse n divided by ORIGINAL divisor1 := div(9, 1)
After
Mobius roundtrip m=9 inverse n divided by ORIGINAL divisor1 = 9
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S062
S064 · implies
Before
Mobius roundtrip m=9 inverse F(n/1):actual positive input=9;check complete integer range1..9
After
ALL positive divisors INCREASING=[1, 3, 9];not only primes/proper divisors or all integers<=n
Conditions: none
Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.
Step identifiers and dependencies
Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S063
S065 · calculation
Before
Mobius roundtrip m=9 inverse F(n/1) f(1) ORIGINAL a times divisor := mul(4, 1)
After
Mobius roundtrip m=9 inverse F(n/1) f(1) ORIGINAL a times divisor = 4
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S064
S066 · calculation
Before
Mobius roundtrip m=9 inverse F(n/1) f(1) plus ORIGINAL b := sum(4, -4)
After
Mobius roundtrip m=9 inverse F(n/1) f(1) plus ORIGINAL b = 0
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S065
S067 · calculation
Before
Mobius roundtrip m=9 inverse F(n/1) f(3) ORIGINAL a times divisor := mul(4, 3)
After
Mobius roundtrip m=9 inverse F(n/1) f(3) ORIGINAL a times divisor = 12
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S066
S068 · calculation
Before
Mobius roundtrip m=9 inverse F(n/1) f(3) plus ORIGINAL b := sum(12, -4)
After
Mobius roundtrip m=9 inverse F(n/1) f(3) plus ORIGINAL b = 8
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S067
S069 · calculation
Before
Mobius roundtrip m=9 inverse F(n/1) f(9) ORIGINAL a times divisor := mul(4, 9)
After
Mobius roundtrip m=9 inverse F(n/1) f(9) ORIGINAL a times divisor = 36
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S068
S070 · calculation
Before
Mobius roundtrip m=9 inverse F(n/1) f(9) plus ORIGINAL b := sum(36, -4)
After
Mobius roundtrip m=9 inverse F(n/1) f(9) plus ORIGINAL b = 32
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S069
S071 · calculation
Before
Mobius roundtrip m=9 inverse F(n/1) full forward divisor sum := sum(0, 8, 32)
After
Mobius roundtrip m=9 inverse F(n/1) full forward divisor sum = 40
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S070
S072 · calculation
Before
Mobius roundtrip m=9 inverse mu(1) times SAME F(n/1) := mul(1, 40)
After
Mobius roundtrip m=9 inverse mu(1) times SAME F(n/1) = 40
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S071
S073 · implies
Before
Mobius roundtrip m=9 inverse Mobius at ORIGINAL divisor3:actual positive input for mu=3
After
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
Conditions: none
Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search.
Step identifiers and dependencies
Rule: mobius_complete_UPF_input_role · Uses: G001, S072
S074 · implies
Before
ORIGINAL positive integer absolute magnitude=3>1
After
initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted
Conditions: none
Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime.
Step identifiers and dependencies
Rule: upf_factorization_remaining_product_invariant · Uses: G001, S073
S075 · calculation
Before
upf canonical complete product factor0 := mul(1, 3)
After
upf canonical complete product factor0 = 3
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S074
S076 · implies
Before
exhausted trial divisors below2;terminal remaining=3;d*d>remaining;ORIGINAL n=3
After
complete increasing prime multiset=[3];full product=3;a composite terminal remainder would have an already tested smaller divisor
Conditions: none
If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors.
Step identifiers and dependencies
Rule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S075
S077 · implies
Before
Mobius roundtrip m=9 inverse Mobius at ORIGINAL divisor3:input=3;COMPLETE prime multiset=[3];distinct primes=[3]
After
mu=-1;empty product case=No;repeated prime=No
Conditions: none
The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.
Step identifiers and dependencies
Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S076
S078 · calculation
Before
Mobius roundtrip m=9 inverse n divided by ORIGINAL divisor3 := div(9, 3)
After
Mobius roundtrip m=9 inverse n divided by ORIGINAL divisor3 = 3
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S077
S079 · implies
Before
Mobius roundtrip m=9 inverse F(n/3):actual positive input=3;check complete integer range1..3
After
ALL positive divisors INCREASING=[1, 3];not only primes/proper divisors or all integers<=n
Conditions: none
Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.
Step identifiers and dependencies
Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S078
S080 · calculation
Before
Mobius roundtrip m=9 inverse F(n/3) f(1) ORIGINAL a times divisor := mul(4, 1)
After
Mobius roundtrip m=9 inverse F(n/3) f(1) ORIGINAL a times divisor = 4
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S079
S081 · calculation
Before
Mobius roundtrip m=9 inverse F(n/3) f(1) plus ORIGINAL b := sum(4, -4)
After
Mobius roundtrip m=9 inverse F(n/3) f(1) plus ORIGINAL b = 0
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S080
S082 · calculation
Before
Mobius roundtrip m=9 inverse F(n/3) f(3) ORIGINAL a times divisor := mul(4, 3)
After
Mobius roundtrip m=9 inverse F(n/3) f(3) ORIGINAL a times divisor = 12
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S081
S083 · calculation
Before
Mobius roundtrip m=9 inverse F(n/3) f(3) plus ORIGINAL b := sum(12, -4)
After
Mobius roundtrip m=9 inverse F(n/3) f(3) plus ORIGINAL b = 8
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S082
S084 · calculation
Before
Mobius roundtrip m=9 inverse F(n/3) full forward divisor sum := sum(0, 8)
After
Mobius roundtrip m=9 inverse F(n/3) full forward divisor sum = 8
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S083
S085 · calculation
Before
Mobius roundtrip m=9 inverse mu(3) times SAME F(n/3) := mul(-1, 8)
After
Mobius roundtrip m=9 inverse mu(3) times SAME F(n/3) = -8
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S084
S086 · implies
Before
Mobius roundtrip m=9 inverse Mobius at ORIGINAL divisor9:actual positive input for mu=9
After
complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search
Conditions: none
Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search.
Step identifiers and dependencies
Rule: mobius_complete_UPF_input_role · Uses: G001, S085
S087 · implies
Before
ORIGINAL positive integer absolute magnitude=9>1
After
initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted
Conditions: none
Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime.
Step identifiers and dependencies
Rule: upf_factorization_remaining_product_invariant · Uses: G001, S086
S088 · implies
Before
next divisor=2;previous remaining=9;all smaller divisors exhausted
After
extracted multiplicity=0;next remaining=9;remaining modulo2=1;extracted multiset=[]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S087
S089 · calculation
Before
upf extract divisor3 occurrence0 := div(9, 3)
After
upf extract divisor3 occurrence0 = 3
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S088
S090 · calculation
Before
upf extract divisor3 occurrence1 := div(3, 3)
After
upf extract divisor3 occurrence1 = 1
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S089
S091 · implies
Before
next divisor=3;previous remaining=9;all smaller divisors exhausted
After
extracted multiplicity=2;next remaining=1;remaining modulo3=1;extracted multiset=[3, 3]
Conditions: none
Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.
Step identifiers and dependencies
Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S090
S092 · calculation
Before
upf canonical complete product factor0 := mul(1, 3)
After
upf canonical complete product factor0 = 3
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S091
S093 · calculation
Before
upf canonical complete product factor1 := mul(3, 3)
After
upf canonical complete product factor1 = 9
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S092
S094 · implies
Before
exhausted trial divisors below4;terminal remaining=1;d*d>remaining;ORIGINAL n=9
After
complete increasing prime multiset=[3, 3];full product=9;a composite terminal remainder would have an already tested smaller divisor
Conditions: none
If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors.
Step identifiers and dependencies
Rule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S093
S095 · implies
Before
Mobius roundtrip m=9 inverse Mobius at ORIGINAL divisor9:input=9;COMPLETE prime multiset=[3, 3];distinct primes=[3]
After
mu=0;empty product case=No;repeated prime=Yes
Conditions: none
The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.
Step identifiers and dependencies
Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S094
S096 · calculation
Before
Mobius roundtrip m=9 inverse n divided by ORIGINAL divisor9 := div(9, 9)
After
Mobius roundtrip m=9 inverse n divided by ORIGINAL divisor9 = 1
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S095
S097 · implies
Before
Mobius roundtrip m=9 inverse F(n/9):actual positive input=1;check complete integer range1..1
After
ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n
Conditions: none
Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.
Step identifiers and dependencies
Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S096
S098 · calculation
Before
Mobius roundtrip m=9 inverse F(n/9) f(1) ORIGINAL a times divisor := mul(4, 1)
After
Mobius roundtrip m=9 inverse F(n/9) f(1) ORIGINAL a times divisor = 4
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S097
S099 · calculation
Before
Mobius roundtrip m=9 inverse F(n/9) f(1) plus ORIGINAL b := sum(4, -4)
After
Mobius roundtrip m=9 inverse F(n/9) f(1) plus ORIGINAL b = 0
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S098
S100 · calculation
Before
Mobius roundtrip m=9 inverse F(n/9) full forward divisor sum := sum(0)
After
Mobius roundtrip m=9 inverse F(n/9) full forward divisor sum = 0
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S099
S101 · calculation
Before
Mobius roundtrip m=9 inverse mu(9) times SAME F(n/9) := mul(0, 0)
After
Mobius roundtrip m=9 inverse mu(9) times SAME F(n/9) = 0
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S100
S102 · calculation
Before
Mobius roundtrip m=9 inverse complete inverse-term sum := sum(40, -8, 0)
After
Mobius roundtrip m=9 inverse complete inverse-term sum = 32
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S101
S103 · calculation
Before
Mobius roundtrip m=9 inverse direct ORIGINAL a times n := mul(4, 9)
After
Mobius roundtrip m=9 inverse direct ORIGINAL a times n = 36
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S102
S104 · calculation
Before
Mobius roundtrip m=9 inverse direct plus ORIGINAL b := sum(36, -4)
After
Mobius roundtrip m=9 inverse direct plus ORIGINAL b = 32
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S103
S105 · implies
Before
Mobius roundtrip m=9 inverse:actual n=9;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 9, 40, 40], [3, -1, 3, 8, -8], [9, 0, 1, 0, 0]]
After
complete reconstructed=32;direct f(n)=32;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)
Conditions: none
Swap the finite sums over d*e|n.The inner Mobius sum is1 for n/e=1 and0 otherwise by pairing distinct-prime subsets,leaving only e=n.For n1,the d1 term and mu1=1 remain;do not mix indices from the alternate equivalent convention.
Step identifiers and dependencies
Rule: mobinv_general_finite_double_sum_and_Mobius_cancellation · Uses: G001, S104